Question

A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is

• A. 1
• B. \(\frac{1}{3}\)
• C. \(\frac{4}{3}\)
• D. \(\frac{2}{3}\)

Hint:

When working with perpendicular lines, remember that the slopes of perpendicular lines are negative reciprocals of each other. If the slope of one line is \( m \), the slope of the perpendicular line will be \( \frac{1}{m} \) with the opposite sign. This property is key to solving problems involving perpendicular lines and finding their equations.

Answer 1

The Correct Option is C

The correct answer is (C) :\(\frac{4}{3}\).

Answer 2

The correct answer is: (C) \(\frac{4}{3}\) .

We are given that a line passes through the point (2, 2) and is perpendicular to the line given by the equation \( 3x + y = 3 \).

To find the y-intercept of this line, follow these steps:

• Step 1: Find the slope of the given line \( 3x + y = 3 \) by rewriting it in slope-intercept form \( y = mx + b \). Rearranging the equation:
  \( y = -3x + 3 \)
  The slope of the given line is \( m = -3 \).
• Step 2: The slope of the line that is perpendicular to this one will be the negative reciprocal of -3, which is \( m' = \frac{1}{3} \).
• Step 3: Now use the point-slope form of the equation to find the equation of the line that passes through the point (2, 2) with slope \( \frac{1}{3} \):
  \( y - y_1 = m(x - x_1) \)
  Substitute \( x_1 = 2, y_1 = 2, m = \frac{1}{3} \):
  \( y - 2 = \frac{1}{3}(x - 2) \)
  Expanding:
  \( y - 2 = \frac{1}{3}x - \frac{2}{3} \)
  \( y = \frac{1}{3}x + \frac{4}{3} \)
• Step 4: The y-intercept of the line is the constant term, which is \( \frac{4}{3} \).

Thus, the y-intercept of the line is \(\frac{4}{3}\) .